Here we briefly recall the notion of Riemannian metric on a Lie groupoid introduced in [1]}. Such a notion of Riemannian metric is compatible with the groupoid composition so that it plays an important role in our work. We start by recalling that a submersion \(\pi :(E,\eta ^E)\rightarrow B\)  with \((E,\eta ^E)\)  a Riemannian manifold is said to be Riemannian if the fibers of it are equidistant (transverse condition). In this case the base \(B\)  inherits a metric \(\eta ^B:=\pi _\ast \eta ^E\)  for which the linear map \(d\pi (e)=(\textnormal {ker}(d\pi (e)))^\perp \rightarrow T_{\pi (e)}B\)  is an isometry for all \(e\in E\) .
